10 million Accounts Puzzle, 1,000 followers problem

Algebra Level 5

Have you ever wondered how brilliant started? Well, here's the story of brilliant from its roots. Be like Sherlock Holmes and read carefully, there are clues to this problem scattered throughout the story, some are quite well hidden. I have bolded one part which may have been too hard to catch. Most of the info in this story is true except the parts about Daniel, Prasun, Brilliant's start date (I changed this for convenience) and the staff incident.

Assume today is May 11st. Exactly 2.5 years ago (Nov. 11th), Calvin, Sue, and Peter got together to form a site to attract the smartest kids in the world, each made his/her own account. Luring them not with rewards, but with intriguing challenges submitted by other users from around the world, Brilliant began to grow. The next day, Daniel, being the genius he was, became the first person to join this new site. He began making thousands of the best problems the all knowing Gods of this universe had ever seen.

Unfortunately, that same day, Prasun, while trolling people on Omeggle, saw an ad for a math website by the title "brilliant.org". He moved most of his trolling focus from Omeggle to Brilliant and the growth rate drastically slowed down.

Also on that very day, Calvin, being the math God he is, correctly hypothesized that Brilliant's growth rate would take the form \(\text{# of accounts}=\lfloor a\cdot 5^{bx}+c\rfloor\) where \(a,b,c\) are constants, \( a,c \) are integers, and \(x\) is the amount of time (measured in months) elapsed from Nov. 11th. Unfortunately, he had to change it when Prasun the troll joined to the new growth rate \(\text{# of accounts}=\lfloor a\cdot e^{bx}+c\rfloor\) where \(e\) is the natural logarithm base.

Many moons passed and brilliant continued to grow at the exponential rate law mentioned immediately above.

One groggy Thursday morning while Peter was dozing off at his desk, Silas burst into the room screaming at the top of his lungs, "We did it! We hit 1 million accounts!". Peter, alarmed, immediately woke up to see a jubilant Silas on the other side of his work cubical. Silas began to run around the room and Aaron soon joined him. David walked into the room after hearing all the commotion only to see a smiling Silas and equally joyful Aaron barreling down the isle straight towards him. BAM! The three collided in what can only be described as "the second Big Bang". After the stars disappeared from the three Stooges' eyes, David realized his phone was broken. Being the physics genius he is, he immediately began to calculate how much force was required to break his phone's 1/32 inch thick anodized aluminum backplate. At the same time, Silas and Aaron promised to buy David a new phone since Christmas was coming up in 14 days.

As days turned to weeks and weeks to months, Brilliant became swamped with even more new members. Sue realized that Brilliant needed more staff members so she posted this note looking for more people to hire. In this note, she stated that Brilliant had 1.4 million accounts. However, this was actually a slight miscount due to technical errors. Today, May 11th, 2015, Brilliant finally reached exactly 1.4 million accounts.

Based on the information given above, in which month and year will Brilliant reach 10 million accounts?

Details and Assumptions:

  • For simplicity's sake, assume all months have 30 days.

  • The floor function is just for "technical correctness" (you can't have 10.5 accounts). You don't need it in your calculations as it won't affect the result (assuming it's correct) by more than a day. And even being 7 days off won't affect your answer.

  • You may use a computer for equation solving. When working with \(b\), I suggest using at the least the ten-thousandths place,.

  • 1 million is the number one followed by 6 zeros.

Note that in reality, my prediction is that it might take a little longer to reach 10 million accounts than the actual answer to this question suggests.

This is my unofficial 1000 followers problem. I don't know how much attention it will get because it's a new type of problem. Tell me how you like it and I might make another one. So please, if I may ask, do take the time to at least read it.

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